# Gear ratio

Gears on a piece of farm equipment, gear ratio 1:1.61

The gear ratio is the relationship between the number of teeth on two gears that are meshed or two sprockets connected with a common roller chain, or the circumferences of two pulleys connected with a drive belt.

In the picture to the right, the smaller gear has thirteen teeth, while the second, larger gear has twenty-one teeth. The gear ratio is therefore 13/21 or 1/1.61 (also written as 1:1.61)

The first number in the ratio is usually the gear that power is applied to. In an automobile the first number is the gear receiving power from the engine.

This means that for every one revolution of the smaller gear, the larger gear has made 1/1.61, or 0.62, revolutions. In practical terms, the larger gear turns more slowly.

Suppose the largest gear in the picture has 42 teeth, the gear ratio between the second and third gear then is; 21/42 = 1/2 and for every revolution of the smallest gear the largest gear has only turned 0.62/2 = 0.31 revolution, a total reduction of around 1:3.

Since the number of teeth is also proportional to the circumference of the gear wheel (the bigger the wheel the more teeth it has) the gear ratio can also be expressed as the relationship between the circumferences of both wheels (where d is the diameter of the smaller wheel and D is the diameter of the larger wheel):

[itex]gr = (\pi \times d) / (\pi \times D) = d/D[itex]

Since the diameter is equal to twice the radius;

[itex]gr = d / D = (2 \times r) /( 2 \times R) = r / R[itex]

as well.

## Example

If the gear ratios of a gearbox in a typical automobile are given as:

GearRatio
1st gear1:17
2nd gear1:10
3rd gear1:6
4th gear1:4
reverse 1:15

and given a tire size of 155x15 with a circumference of +/- 2 m, it can be calculated what the speed of the car will be for every 1000 rpm (1000 revolutions per minute) of the engine:

1st gear: every 1000 rpm of the engine = 1000/17 = 58.8 rpm of the wheel = 58.8 x 2 = 117.6 m/min distance traveled = 117.6/1000 x 60= 7 km/h
2nd gear: 1000/10 x 2 /1000 x 60 = 12 km/h
3rd gear: 1000/6 x 2 /1000 x 60 = 20 km/h
4th gear: 1000/4 x 2 /1000 x 60 = 30 km/h

If the maximum rpm of the engine is 5000, the maximum speed of this car in 4th gear will be 5000/1000 x 30 = 150 km/h, assuming the engine has sufficient power to overcome drag.

Modern automobiles often have a 5th gear with a 1:1 gear ratio, where the wheels turn with the same rpm as the engine. Some cars even have an overdrive with a 1.2:1 gear ratio, where the wheels turn faster than the engine.

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